JP6760739B2 - Information processing equipment, information processing methods and programs - Google Patents

Description

The present invention relates to an information processing device, an information processing method and a program.

Combinatorial optimization problems include combinations that minimize the sum of costs among subset combinations that include each element of a given set once, such as crew scheduling decision processing and delivery plan decision processing. There is a set partitioning problem to find.
However, in the partition of a set problem, as the number of subset candidates increases, the amount of calculation increases significantly, and it becomes difficult to calculate the solution within the required time.
There is a method of Patent Document 1 as a method of calculating a solution of a combination optimization problem such as a set partitioning problem.

Japanese Unexamined Patent Publication No. 2001-188984

However, even in Patent Document 1, when the scale of the problem becomes large, it takes time to calculate, and it is difficult to obtain a highly accurate solution within the required time.
On the other hand, an Ising model calculation device has been developed that can efficiently calculate the spin state that minimizes the Hamiltonian in the Ising model.
The Ising model is a simple model that calculates the spin directions of the atoms that make up a crystal. It is assumed that there are atoms at the lattice points on the lattice, and each atom has a spin that takes either upwards or downwards. The Hamiltonian of the Ising model is defined by taking advantage of the phenomenon that the spins of adjacent atoms want to be aligned.
By expressing the combination optimization problem as an Ising model, it is possible to efficiently solve the combination optimization problem using the Ising model calculation device.
However, the format of the equation expressing the partition of a set problem and the Hamiltonian equation of the Ising model are different, and the partition of a set problem cannot be expressed as the Ising model.
An object of the present invention is to make it possible to efficiently find a solution to a partition of a set problem.

The information processing apparatus of the present invention is an acquisition means for acquiring information on a plurality of subsets S ₁ , S ₂ , ... As candidate information which is information on candidates for subsets in a partitioning problem that divides a set M. , The set D _k of the number j of the subset S _j including _k is acquired for each element k of the set M based on the candidate information acquired by the acquisition means , and is the number of elements of the acquired D _k. Obtain d _k , obtain P _k which is a set of pairs of elements of D _k , determine the interaction coefficient J _ij corresponding to the element of P _k as -1, and determine the external magnetic field corresponding to the element of D _k. By determining the variable h _j as − (d _k -2) and applying D _k , P _k , J _ij, and h _j to a predetermined equation as a local Hamiltonian equation for k, the local for k such a formula H _k Hamiltonian, seeking formula H _k the values of variables other than the variable σ representing the spin direction in the Ising model is determined, the H _k obtained for each element k of the set M, previously It has a generation means for generating information indicating Hamiltonian's equation in the Zing model corresponding to the partition of a set problem by adding all the elements of M using the defined equation .
By having such a configuration, it is possible to generate information showing the Hamiltonian equation of the Ising model representing the partitioning problem, so it is possible to apply the Ising model calculation device to efficiently find the solution of the partitioning problem. Can be.

According to the present invention, it is possible to efficiently find a solution to the partition of a set problem.

FIG. 1 is a diagram showing an example of a system configuration of a calculation system. FIG. 2 is a diagram showing an example of the hardware configuration of the information processing device. FIG. 3 is a diagram showing an example of the functional configuration of the information processing device. FIG. 4 is a flowchart showing an example of processing of the information processing apparatus. FIG. 5 is a diagram illustrating an outline of the partition of a set problem. FIG. 6 is a diagram showing the relationship between the constraint condition and the Hamiltonian. FIG. 7 is a flowchart showing an example of the conversion process. FIG. 8 is a diagram illustrating an example of implementation of an Ising model calculation device. FIG. 9 is a diagram illustrating an example of implementation of an Ising model calculation device.

<Embodiment 1>
Hereinafter, embodiments of the present invention will be described with reference to the drawings.
FIG. 1 is a diagram showing an example of a system configuration of a calculation system that performs a process of calculating a set partition problem of the present embodiment. The calculation system includes an information processing device 100 and an Ising model calculation device 101.
The information processing device 100 is an information processing device such as a personal computer (PC) or a server device that acquires information on candidates for subsets of the partitioning problem and converts the partitioning problem into the Hamiltonian format of the Ising model.

The Ising model calculation device 101 is a device that calculates the spin state that minimizes the Hamiltonian for the specified Ising model. The Ising model calculation device 101 is, for example, a device having a spin circuit for each spin in the Ising model. Each spin circuit is a memory element that stores the state of each spin (upward or downward, +1 or -1), a memory element that stores the interaction coefficient between spins adjacent to its own spin, and the Ising model Hamiltonian. It has a memory element that stores a value of a coefficient corresponding to an external magnetic field coefficient indicating the magnitude of the influence of an external magnetic field (hereinafter, simply referred to as an external magnetic field coefficient for simplification of description).
In addition, each spin circuit has a logic circuit used to determine the next-generation spin state of each spin. The generation is an index for indicating how long a period has passed since the initial state, with the initial state as the first generation, and increases by one each time the set clock elapses. For example, the state in which the clock set from the A generation has passed is the A + 1 generation.
The Ising model calculation device 101 determines the spin state of the next generation based on the spin state of the current generation. Each spin circuit notifies its own spin state to the spin circuits of adjacent spins, and acquires the spin states of adjacent spins from the spin circuits of adjacent spins. In addition, each spin circuit acquires an interaction coefficient between a memory element and an adjacent spin and an external magnetic field coefficient. Each spin circuit updates its own spin state based on the acquired adjacent spin states, interaction coefficients, and external magnetic field coefficients.
Each spin circuit attempts to update its spin state in the same direction as the adjacent spin with a positive interaction coefficient and in the opposite direction to the adjacent spin with a negative interaction coefficient. Each spin circuit determines the state of its new spin by a majority vote of the effects from adjacent spins. As a result, the Ising model calculation device 101 changes each spin in the Ising model in the direction in which the energy of the Ising model decreases.
The information processing apparatus 100 acquires a candidate for a subset of the set partitioning problem, and based on the acquired candidate information, acquires an equation obtained by converting the set partitioning problem into the Hamiltonian format of the Ising model. Then, the information processing apparatus 100 transmits the acquired information of the Hamiltonian equation to the Ising model calculation device 101. The Ising model calculation device 101 calculates the direction of each spin that minimizes the transmitted Hamiltonian equation, and transmits the calculated direction of each spin to the information processing device 100. The information processing apparatus 100 determines a subset that will be the final division result from the acquired subset candidates based on the direction of each transmitted spin.

FIG. 2 is a diagram showing an example of the hardware configuration of the information processing device 100. The information processing device 100 includes a CPU 201, a main storage device 202, an auxiliary storage device 203, input / output I / F 204, and a device I / F 205. The CPU 201, the main storage device 202, the auxiliary storage device 203, the input / output I / F 204, and the device I / F 205 are connected to each other via the system bus 206.
The CPU 201 is a central processing unit that controls the processing of the information processing device 100. The main storage device 202 is a storage device that functions as the main memory, work area, and the like of the CPU 201. The auxiliary storage device 203 is a storage device that stores various programs, various setting information, and the like. The input / output I / F 204 is an interface used when receiving input from an input device such as a mouse, keyboard, or touch panel, or outputting a calculation result or the like to an output device such as a monitor or touch panel. The device I / F205 is an interface used when connecting to an external device such as an Ising model calculation device.
When the CPU 201 executes the process based on the program stored in the auxiliary storage device 203 or the like, the function of the information processing device 100 described later in FIG. 3 and the process of the flowchart described later in FIGS. 4 and 7 are realized.

FIG. 3 is a diagram showing an example of the functional configuration of the information processing apparatus 100. The information processing device 100 includes a candidate information acquisition unit 301, a conversion unit 302, a device control unit 303, and a determination unit 304.
The candidate information acquisition unit 301 acquires the candidate information of the subset in the set partitioning problem. For example, the candidate information acquisition unit 301 acquires the candidate information of the subset in the set partitioning problem which is the calculation target stored in advance in the auxiliary storage device 203 from the auxiliary storage device 203. Further, the candidate information acquisition unit 301 may acquire the candidate information of the subset in the set partitioning problem to be calculated from an external storage device such as an external HDD or an external USB memory.
Further, the candidate information acquisition unit 301 may acquire the candidate information selected based on the input data selection screen displayed on the display device or the like or the operation of the user via the input device from the auxiliary storage device 203 or the like. Good. Further, the candidate information acquisition unit 301 may acquire the candidate information of the subset in the set partitioning problem to be calculated based on the input screen displayed on the display device or the like or the user's operation via the input device. Good. Further, the candidate information acquisition unit 301 may acquire the candidate information of the subset in the set partitioning problem to be calculated from an external server or the like connected via the network. The candidate information acquisition unit 301 passes the acquired candidate information to the conversion unit 302.

The conversion unit 302 converts the set partitioning problem of the calculation target into the Ising model based on the candidate information acquired from the candidate information acquisition unit 301. The set partitioning problem is a problem of partitioning a set to be divided under the constraint condition that there is only one subset including the element k of the set to be divided. The conversion unit 302 converts the set partition problem into the Ising model by generating a Hamiltonian equation of the Ising model that satisfies the constraint condition of the set partition problem when the minimum value is taken.
The device control unit 303 transmits the information of the Hamiltonian equation of the Ising model generated by the conversion unit 302 to the Ising model calculation device 101, so that the transmitted equation is the minimum value to the Ising model calculation device 101. Instruct to calculate the state of the spin. The Ising model calculation device 101 calculates the spin state in which the value of the received Hamiltonian formula is the minimum value, and transmits the calculation result to the device control unit 303. The device control unit 303 passes the calculation result received from the Ising model calculation device 101 to the determination unit 304.
The determination unit 304 is the final candidate for the division result acquired by the candidate information acquisition unit 301 based on the spin state in which the value of the Hamiltonian equation passed from the device control unit 303 is the minimum value. Determine the subset that will be the result of the split.

FIG. 4 is a flowchart showing an example of processing of the information processing apparatus 100. In the present embodiment, the calculation target is a set partitioning problem that divides a set M = {1, 2, ..., M} composed of m elements.
In S401, the candidate information acquisition unit 301 acquires the candidate information of the subset in the set partitioning problem stored in advance in the auxiliary storage device 203 from the auxiliary storage device 203.
The auxiliary storage device 203 is, for example, information on an array (hereinafter referred to as a candidate array) having the same number of elements as the number of elements of the set to be divided as candidate information which is information on the candidates of the subset in the set partition problem. Is memorized in advance. The candidate array is an array showing one candidate of a subset in the set partitioning problem, and when the number of elements of the set M to be partitioned is m, it is a one-dimensional array having m elements. The k (1 ≦ k ≦ m) th element of the candidate array corresponds to the kth element of the set M to be divided. Each element of the candidate array takes a value of either 0 or 1, and the value of the k-th element of the candidate array is 1, which means that the subset indicated by the candidate array includes the k-th element of the set M. Indicates that Further, when the value of the k-th element of the candidate array is 0, it means that the subset indicated by the candidate array does not include the k-th element of the set M. Further, the candidate information is not limited to an array as long as it is information indicating which element is included in the elements of the set to be divided. For example, which element of the set to be divided is used. It may be text data or the like indicating whether or not it is included.
The candidate information acquisition unit 301 acquires, for example, n candidate sequences corresponding to n different subsets S _j (1 ≦ j ≦ n) from the auxiliary storage device 203. Each of the acquired n candidate arrays is a one-dimensional array having m elements. Further, the information processing apparatus 100 can grasp the number of elements of the set to be divided in the set partitioning problem of the calculation target by acquiring the number of elements of the acquired candidate array.

In S402, the conversion unit 302 generates a Hamiltonian in the Ising model showing the set partitioning problem of the calculation target based on the candidate information acquired in S401. As a result, the conversion unit 302 can convert the set partitioning problem of the calculation target into a problem of finding the spin state when the Hamiltonian in the Ising model becomes the minimum value. Details of the processing of S402 will be described later in FIG.
In the following, the mechanism for converting the set partitioning problem into the Ising model will be described. The partition of a set problem is the following problem. Given a set M consisting of m elements = {1, 2, ..., m}, n subsets S _j and their costs c _j , all the elements k ∈ M of the set M The problem is to find the one that minimizes the sum of costs among the combinations of subsets that are included exactly once. It becomes 1 if element k of the set M is included in the S _j, k defines the coefficients a _kj such that 0 is not included in the S _j. Then, 1 becomes when the subset S _j is selected, the 0 and becomes variable when the subset S _j is not selected when the x _j, set partitioning problem can be formulated in Equation 1 below. The variable x _j can be regarded as a variable indicating whether or not the subset S _j is selected as the division result of the set M to be divided. The uppermost equation of Equation 1 is the objective function in the partition of a set problem.

FIG. 5 is a diagram illustrating an outline of the partition of a set problem. FIG. 5 shows S401, in which nine subsets S _j (1 ≦ j ≦ 9), that is, S ₁ , S, are candidates for the subset of the set {1, 2, 3, ..., 7} to be divided. It is a figure which shows the example of the coefficient a _{k j} when the candidate information of ₂ , ..., S ₉ is acquired.
Row k of the table in FIG. 5 indicates the element k of the set to be divided. Then, column j of the table of FIG. 5 shows each subset S _j . The shaded elements of each column in the table of FIG. 5 indicate that the subset corresponding to that column contains the elements of the corresponding set. The set partitioning problem in FIG. 5 is a problem of selecting a subset from S _j so that each element of the set {1, 2, 3, ..., 7} appears exactly once. it can. In the example of FIG. 5, the correct answer is a combination of S ₁ , S ₇ and S ₉ , or a combination of S ₄ and S ₈ .

In this embodiment, the information processing apparatus 100 converts the set partitioning problem into the Hamiltonian equation of the Ising model. The Ising model calculation device 101 then calculates the spin state that minimizes the transformed Hamiltonian. That is, the Ising model calculation device 101 calculates the spin state that minimizes the Hamiltonian. If the partition of a set problem to be calculated has an objective function, the Ising model computing device 101 must calculate the spin state that minimizes other objective functions in addition to the Hamiltonian. In this case, the Ising model calculation device 101 requires more hardware resources (for example, a memory element in each spin circuit) than when calculating the spin state that minimizes the Hamiltonian. Therefore, the Ising model calculation device 101 may not be able to calculate the spin state that minimizes other objective functions in addition to the Hamiltonian due to lack of hardware resources.
Further, in the partition of a set problem, even if it is not the optimum solution that satisfies the objective function, it may be sufficient if a local solution is obtained. Therefore, in the present embodiment, the calculation system calculates the set partitioning problem without the objective function. Thereby, the calculation system can save resources such as hardware resources in the Ising model calculation device 101.
The partition of a set problem without an objective function can be formulated as in Equation 2 below.

The format of Equation 2 is different from the Hamiltonian format of the Ising model described later in Equation 5. Therefore, the Ising model calculation device 101 cannot calculate the set partitioning problem represented by the equation 2.
The variable x _j in Equation 2 is changed to the variable σ _j (for example, 1 when the spin is upward and -1 when the spin is downward) indicating the spin direction of the atom, as shown in Equation 3 below. If it is changed, it can be converted into the formula of the following formula 4. Variable x _j is selected, since the subset S _j is a variable indicating whether it is selected as the division result of the set M of division target as the division result of the variable sigma _j be the subset S _j is the set of split target M It can be regarded as a variable indicating whether or not it has been done. That is, when σ _j is 1, it means that S _j was selected as the result of dividing M, and when σ _j is -1, it means that S _j was not selected as the result of dividing M. It is a variable to show.

The Hamiltonian of the Ising model will be described. Hamiltonian is a physical quantity corresponding to energy. The Hamiltonian of the Ising model can be expressed as the following equation 5. The first term on the right side of Equation 5 is between the spins of i and the spins of j and the spins of i and the spins of j for all combinations of spins (i, j) connected on the grid. It is the sum of the products of the interaction coefficient J _ij , which indicates the magnitude of the interaction of. The second term on the right side of Equation 5 is the potential energy due to the external magnetic field. The external magnetic field coefficient h _j is a coefficient indicating the magnitude of the influence of the external magnetic field.

The elements k of the set M of division target and a set of a _kj = 1 and becomes j and D _k, the set D _k can be expressed as the following equation 6. The set D _k can be regarded as the set of the numbers j of the subset S _j including the element k.

Let d _k be the number of elements in the set D _k . Further, assuming that the set of the pair (i, j) of the elements of the set D _k is P _k , the set P _k can be expressed by the following equation 7.

Assuming that the local Hamiltonian equation for the element k is H _k , the equation H _{k is set} to the following equation 8 so that when the equation H _k has the minimum value, there is only one subset containing the element k. The constraint of existence will be met.

Assuming that the interaction coefficient J _ij in the Hamiltonian of the Ising model is -1 and the external magnetic field coefficient h _j is-(d _k -2), H _k of Equation 8 is the Hamiltonian of the Ising model as shown in Equation 9 below. It becomes the formula of.

The reason why the constraint condition that there is only one subset including the element k is satisfied when the equation H _k shown in the equation 8 has the minimum value will be described.
_{Assuming that the} number of sigma _j with a value of 1 is e (0 ≦ e ≦ d _k ) and the number of sigma with a value of -1 is (d _{k −} e), the set D _k The sum of σ _j of is as shown in Equation 10 below.

The number of elements of P _k , which is a set of pairs (i, j) of elements included in D _k , is d _k (d _k -1) / 2. The number of pairs (i, j) in which σ _i σ _j = -1 among the elements of P _k is e (d _{k −} e). The number of pairs (i, j) in which σ _i σ _j = 1 among the elements of P _k is d _k (d _k -1) / 2-e (d _k −e).
Therefore, the added value for all the pairs (i, j) included in P _k is as shown in Equation 11 below.

Here, Equation 8 can be transformed and expanded as Equation 11 + Equation 10 × (d _k- 2). This equation transformation and expansion is shown in Equation 12.

As shown in Equation 12, Equation ^{8, 2 (e-1) 2} - (d k 2 -3d k +4) / 2 that can be transformed to formula. Since the first term of the modified equation is the square of a real number, it is always 0 or more, and when e = 1, the value is 0. Further, the second term of the modified equation can be regarded as a constant because d _k is a constant. Therefore, the value of Equation 8 takes the minimum value when e is 1. When e is 1, it means that exactly one subset of the subsets including the element k is selected as the division result of the set M. That is, when H _k takes the minimum value, the constraint condition that there is only one subset containing the element k is satisfied.
FIG. 6 is a diagram showing the relationship between the constraint condition and the Hamiltonian. The table of FIG. 6 shows various combinations (16 combinations) of the spin states σ _j (0 ≦ j ≦ 4) (upward or downward, 1 or -1), taking the case of d _k = 4 as an example. It is a table that lists the data. Item 601 is an item indicating whether or not the constraint condition that there is only one spin having an upward (1) value is satisfied. For example, in the data in the second row of the table of FIG. 6, of σ _j (0 ≦ j ≦ 4), only σ ₄ is 1, which satisfies the constraint condition, and the value of item 601 is True. .. Further, for example, in the data in the first row of the table of FIG. 6, all of σ _j (0 ≦ j ≦ 4) are -1, and the constraint condition is not satisfied, and the value of item 601 is False. There is.
Item 602 in the table of FIG. 6 shows what the value of Equation 8 which is the local Hamiltonian equation for element k is. In the example of FIG. 6, the value of Equation 8 is shown to be the minimum at -4. Here, when the values of item 601 and item 602 are compared, it can be seen that the constraint condition is satisfied when the value of Equation 8 is the minimum. That is, it can be confirmed that Equation 8 should be minimized in order to find a solution in which exactly one spin is upward.

FIG. 7 is a flowchart showing an example of a conversion process for converting a set partitioning problem into an Ising model. The details of the processing of S402 will be described with reference to FIG. 7.
In S701, the conversion unit 302 initializes the parameter kp for indicating the element k included in the set M. More specifically, the conversion unit 302 determines the value of kp stored in the main storage device 202 to 1. A kp having a value of 1 indicates an element 1 (= k) included in the set M.
In S702, the conversion unit 302 determines whether or not the value of kp is m or less, which is the number of elements of the set M. When the conversion unit 302 determines that the value of kp is m or less, it proceeds to the process of S703, and when it determines that the value of kp is larger than m, which is the number of elements of the set M, it proceeds to the process of S709.
In S703, the conversion unit 302 specifies a subset including the element k indicated by kp in the subset S _j corresponding to the candidate information acquired in S401, and is a set of the number (j) of the specified subset. Get a set D _k . For example, when a candidate array is acquired as candidate information in S401, the conversion unit 302 uses the candidate array as the candidate array when the value of the kp value element of the candidate array is 1. The corresponding subset is specified as a subset containing the element k.
In S704, the conversion unit 302 acquires the number of elements d _k of the set D _k acquired in S702. For example, when D _k is array structure data, the conversion unit 302 acquires d _k by acquiring the number of elements of D _k acquired in S703.
In S705, the conversion unit 302 acquires a set P _k of pairs of elements included in the set D _k acquired in S702. For example, the conversion unit 302 selects the first element from D _k , selects the second element from the remaining elements, and determines the selected elements as a pair. Further, the conversion unit 302 acquires the set P _k by repeating this process until all the pairs are selected.
In S706, the conversion unit 302 determines the interaction coefficient J _ij and the magnetic field coefficient h _j in the Hamiltonian equation of the Ising model as shown in Equation 5. The conversion unit 302 determines the value of J _ij as -1. Further, the conversion unit 302 determines the value of h _j to be − (d _k − 2) based on the d _k acquired in S704.

In S707, the conversion unit 302 generates a Hamiltonian H _k for the element k of M indicated by kp. For example, it is assumed that the auxiliary storage device 203 stores the information of the equation 9 in the form of H _k in advance. The first term Σ on the right side of Equation 9 means that the sum of J _ij σ _i σ _j is taken for all the elements (pairs of elements of D _k ) included in the set P _k . Further, Σ of the second term on the right side of Equation 9 means that the sum of h _j σ _j is taken for all the elements included in the set D _k .
H _{k in} Equation 9 is an equation used by the Ising model calculation device 101 or the like to calculate the value of σ _j (1 ≦ j ≦ d _k ) at which H _k has the minimum value. Therefore, it is necessary to determine the values of variables (P _k , J _ij , h _j , D _k ) other than σ in Equation 9. In the present embodiment, the conversion unit 302 generates a Hamiltonian for the element k by determining the values of variables (P _k , J _ij , h _j , D _k ) other than σ in Equation 9. To do.
The conversion unit 302 determines the value of P _k in Equation 9 as the set P _k acquired in S705. The conversion unit 302 determines the value of D _{k in} the equation 9 as the set D _k acquired in S703. Further, the conversion unit 302 determines the value of J _{ij in} the equation 9 as J _ij (-1) determined in S706. Further, the conversion unit 302 determines the value of h _{j in} the equation 9 as h _j (− (d _k -2)) determined in S706. Then, the conversion unit 302 generates a local Hamiltonian equation for the element k by applying the determined variables (P _k , J _ij , h _j , D _k ) to the equation 9. The conversion unit 302 stores the generated local Hamiltonian equation information in the main storage device 202.

In S708, the conversion unit 302 adds 1 to the value of kp and then proceeds to the process of S702.
In S709, the conversion unit 302 adds the Hamiltonian H _k of the local Ising model for each element k of the set M generated in S707 for each element k of the set to be divided by using Equation 13, and the entire Ising. Generate information about the Hamiltonian H equation for the model. In this way, the conversion unit 302 can convert the set partition problem into the Ising model by generating the Hamiltonian equation of the Ising model corresponding to the set partition problem. That is, the Ising model corresponding to H is the Ising model corresponding to the partition of a set problem.

Returning to the description of the flowchart of FIG.
In S403, the device control unit 303 transmits the information of the Hamiltonian H equation generated in S402 to the Ising model calculation device 101. By transmitting the information of H, the device control unit 303 instructs the Ising model calculation device to calculate the state (σ _{1 to} σ _n ) of each spin when H takes the minimum value.
The Ising model calculation device 101 calculates the state of each spin of the Ising model corresponding to H when H takes the minimum value, based on the information of H transmitted in S403. For example, the Ising model calculation device 101 prepares a spin circuit for each spin (σ _{1 to} σ _n ) of the Ising model corresponding to H. Then, the Ising model calculation device 101 sets an initial spin state value, an interaction coefficient, and an external magnetic field coefficient in each spin circuit so as to represent the Ising model corresponding to H. The Ising model calculation device 101 randomly sets a value of 1 or -1 as the initial spin state value of each spin circuit. For example, the Ising model calculation device 101 generates a random number, and randomly determines whether the initial spin state value of each spin circuit is 1 or -1 based on the generated random number. After that, each spin circuit changes the state of each spin so that the value of H is the minimum value. When a set number of generations have passed from the first generation, the Ising model calculation device 101 transmits information indicating the state of each spin in that generation to the information processing device 100.

FIG. 8 is a diagram illustrating an example of implementation of the Ising model calculation device 101. An example in which the Ising model calculation device 101 implements each spin circuit will be described with reference to FIG. In FIG. 8, the set M = {1}, D _k = {1, 2, 3, 4}, d _k = 4, P _k = {(1, 2), (1, 3), (1, 4). , (2, 3), (2, 4), (3, 4)}, which is a diagram illustrating an example of an Ising model corresponding to the Hamiltonian of Equation 13. The Ising model of FIG. 8 is an Ising model in which two layers of spins arranged in a grid pattern are overlapped. The Ising model calculation device 101 implements the Ising model by, for example, storing setting information corresponding to each spin (each grid point in FIG. 8) in the Ising model in a memory element of an internal spin circuit. First, the Ising model calculation device 101 determines the interaction coefficient and the external magnetic field coefficient of each spin to all zeros. In this state, each spin is unaffected by the surrounding spins.
Each row in the lower layer of the Ising model of FIG. 8 corresponds to the spin σ _j corresponding to the element j contained in D _k . The Ising model calculation device 101 can make the directions of all spins in the same row in the lower layer the same by determining the interaction coefficient of each spin adjacent to each other in the same row in the lower layer to +1. Thereby, all the spins in the lower row i can be similarly regarded as indicating the spin state of σ _i (whether or not the subset S _i is selected as the division result).

The Ising model calculation device 101 connects the rows of the lower layer to each other by using the upper layer. When the Ising model calculation device 101 connects σ ₁ and σ ₂ , the interaction coefficient between the leftmost spin of the first row of the lower layer and the leftmost spin of the first row of the upper layer is determined to be 1, and the lower layer 2 The interaction coefficient between the spin at the left end of the second row and the spin at the left end of the second upper layer is determined to be 1. Then, the Ising model calculation device 101 determines the interaction coefficient between the spin at the left end of the first row of the upper layer and the spin at the left end of the second row of the upper layer to -1. As a result, the Ising model calculation device 101 can make the connection between the row of σ _{1 and} the row of σ ₂ similar to the connection with the interaction coefficient -1. Similarly, the Zing model calculation device 101 uses a row of σ _{1 and} a row of σ _3, a row of σ _{1 and} a row of σ _4, a row of σ _{2 and} a row of σ ₃ , and a row of σ ₂ . and a sigma ₄ rows, and a row of sigma ₃ rows and sigma _4, connects to the interaction coefficient -1.
In the example of FIG. 8, since d _k = 4, the external magnetic field coefficient h _j is − (d _k -2) = − (4-2) = -2. Therefore, the Ising model calculation device 101 determines the external magnetic field coefficients of the two spins to -1 for each row in the lower layer in order to represent the external magnetic field coefficient -2.
Through the above processing, the Ising model calculation device 101 implements the Ising model corresponding to the designated Hamiltonian. Then, the Ising model calculation device 101 calculates the state of each spin with the Hamiltonian value as the minimum value. The Ising model calculation device 101 acquires the spin state value of each row in the lower layer after the set generation has elapsed from the start of the calculation. The spin state value of each row in the lower layer indicates whether or not the subset corresponding to each row is selected as the split result. The Ising model calculation device 101 transmits the acquired spin state value information to the information processing device 100.
Further, the information processing apparatus 100 may acquire the calculation result by the Ising model calculation device 101 as follows. That is, when the set period elapses from the time when the Ising model calculation device 101 is instructed to perform the calculation, the device control unit 303 sets the spin state value of each row in the lower layer of the Ising model to the Ising model calculation device 101. It may be requested. For example, the Ising model calculation device 101 does not continue to calculate the state of each spin with the Hamiltonian value as the minimum value from the start of calculation until the set generation elapses, but receives a request from the information processing device 100. Continue to calculate until you do. Then, when the Ising model calculation device 101 receives a request from the information processing device, it acquires the spin state value of each row in the lower layer of the Ising model and transmits the acquired state value information to the information processing device 100. Further, the Ising model calculation device 101 may continue to calculate the state of each spin having the Hamiltonian value as the minimum value even after receiving the request from the information processing device 100. In that case, when the Ising model calculation device 101 receives the request for the calculation result again from the information processing device, the Ising model calculation device 101 acquires the spin state value of each row in the lower layer of the Ising model, and the information processing device 100 obtains the acquired state value information. Send to.

In FIG. 9, the set M = {1, 2, 3, ..., M}, there are n subset candidates, and there are m local Hamiltonians of equation 9 for each element of M. It is a figure explaining the outline of the Ising model corresponding to the Hamiltonian of the case formula 13. In the example of FIG. 9, it is assumed that the local Hamiltonian equation for element 1 of M is similar to the Hamiltonian equation in FIG.
In the example of FIG. 9, it is assumed that the Ising model corresponding to the entire Hamiltonian including a plurality of local Hamiltonians is an Ising model in which two layers of spins arranged in a grid pattern are overlapped as in FIG. FIG. 9 is a diagram showing a bird's-eye view of the two-layer Ising model corresponding to the entire Hamiltonian from above.
In each row of the lower layer, each spin is connected with an interaction coefficient of 1, as in FIG. Therefore, all spins in each row in the lower layer have the same state value. The shaded area H ₁ is the area where the part corresponding to the local Hamiltonian H ₁ for element 1 is mounted. The content of the implementation is the same as that in FIG. Region H ₂ is the region where the part corresponding to the local Hamiltonian H ₂ for element 2 is implemented. Further, the region H _m is an region in which the part corresponding to the local Hamiltonian H _m for the element m is implemented. The Ising model calculation device 101 is mounted so that the regions (H ₁ , H ₂ , ..., H _m ) do not overlap. Implementation of the process of the region other than the region H ₁ is similar to the area H _1, i.e., the same as in FIG. The Ising model calculation device 101 implements the Ising model corresponding to the entire Hamiltonian as an Ising model as shown in FIG. 9, and calculates the state of each spin in which the total Hamiltonian is the minimum value. The Ising model calculation device 101 acquires the spin state value of each row in the lower layer after the set generation has elapsed from the start of the calculation. The spin state value of each row in the lower layer indicates whether or not the subset corresponding to each row is selected as the split result. The Ising model calculation device 101 transmits the acquired spin state value information to the information processing device 100.

In S404, the device control unit 303 receives information indicating the state of each spin of the Ising model, which is the calculation result instructed in S403 from the Ising model calculation device 101. Information obtained in step S404, sigma _j corresponding to each subset S _j is information indicating which -1 or 1. The determination unit 304 determines the subset S _j corresponding to σ _j having a value of 1 as being selected as the division result of the set M. Further, the determination unit 304 determines the subset S _j corresponding to σ _j having a value of -1 as not being selected as the division result of the set M.
In S405, the determination unit 304 outputs the information of the subset S _j determined as the division result of the set M in S404. For example, the determination unit 304 displays a confirmation screen for displaying all the subsets S _j determined as the division result of the set M on the display unit of the information processing apparatus 100. By visually recognizing the displayed confirmation screen, the user can determine whether or not the subset in which the division result of the set M is displayed is acceptable. Further, the determination unit 304 may store the information of the subset S _j determined as the division result of the set M in the storage unit of the information processing apparatus 100 as an electronic file. The auxiliary storage device 203 is an example of a storage unit of the information processing device 100. Further, the determination unit 304 may output the information of the subset S _j determined as the division result of the set M by transmitting it to an external device.

As described above, by the processing of the present embodiment, the calculation system can convert the set partition problem into the Ising model problem by generating the Hamiltonian equation in the Ising model corresponding to the set partition problem. As a result, the calculation system can efficiently find the solution of the partition of a set problem via the Ising model calculation device 101.

<Other Embodiments>
Although the preferred embodiment of the present invention has been described in detail above, the present invention is not limited to the specific embodiment.
For example, a part or all of the functional configuration of the above-mentioned calculation system may be implemented in the information processing apparatus 100 as hardware.

100 Information processing device 101 Ising model calculation device 302 Conversion unit

Claims (6)

An acquisition means for acquiring information on a plurality of subsets S ₁ , S ₂ , ... As candidate information that is information on candidates for subsets in the partitioning problem that divides the set M.
Based on the candidate information acquired by the acquisition means, for each element k of the set M, the set D _k of the number j of the subset S _j including _k is acquired, and d is the number of elements of the acquired D _{k. Get k} , get P _k , which is a set of pairs of elements of D _k , determine the interaction coefficient J _ij corresponding to the element of P _k as -1, and the external magnetic field coefficient corresponding to the element of D _k. By determining h _j as − (d _k -2) and applying D _k , P _k , J _ij, and h _j to Equation 1, which is predetermined as the local Hamiltonian equation for k, the local for k. such a formula H _k Hamiltonian, seeking formula H _k the values of variables other than the variable σ representing the spin direction in the Ising model is determined, the H _k obtained for each element k of the set M, previously By adding all the elements of M using the defined equation 2, a generation means for generating information indicating the Hamiltonian equation in the Ising model corresponding to the set division problem, and a generation means.
Information processing device with.

Information indicating the Hamiltonian equation in the Ising model corresponding to the partitioning problem generated by the generation means is transmitted to the Ising model calculation device, and the spin state in which the value of the equation is minimized is transmitted to the Ising model calculation device. Instruction means to instruct the calculation and
A determination means for receiving the information of the calculation result according to the instruction by the instruction means from the Ising model calculation device and determining the division result of the set division problem based on the received information of the calculation result.
The information processing apparatus according to claim 1 , further comprising. The information processing apparatus according to claim 2 , further comprising an output means for outputting the division result of the set partitioning problem determined by the determination means. The information processing apparatus according to claim 2 or 3 , further comprising the Ising model calculation device. It is an information processing method executed by an information processing device.
As candidate information that is information on candidates for subsets in the partitioning problem that divides set M, a plurality of subsets S ₁ , S ₂ The acquisition step to acquire the information of, ...
For each element k of the set M, a subset S containing k, based on the candidate information acquired in the acquisition step. _j Set D of number j _k And acquired D _k The number of elements of d _k And D _k P, which is a set of pairs of elements of _k And P _k Interaction coefficient J corresponding to the element of _ij Is determined to be -1, and D _k External magnetic field coefficient h corresponding to the element of _j -(D _k -2) was decided, and D _k And P _k And J _ij And h _j By applying to equation 1 predetermined as the local Hamiltonian equation for k, the local Hamiltonian equation H for k _k The equation H in which the values of variables other than the variable σ representing the spin direction in the Ising model are determined. _k Was obtained, and H was obtained for each element k of the set M. _k To generate information indicating the Hamiltonian equation in the Ising model corresponding to the partition of a set problem by adding all the elements of M using the predetermined equation 2.
Information processing methods including.

A program for causing a computer to function as each means of the information processing apparatus according to any one of claims 1 to 4.

Images